Theorem elmaprdOLD 8853

Description: Obsolete version of elmaprd 8852 as of 30-Aug-2026. (Contributed by Thierry Arnoux, 13-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
elmaprdOLD.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
elmaprdOLD.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)
elmaprdOLD.3	⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m 𝐴))

Assertion

Ref	Expression
elmaprdOLD	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Proof of Theorem elmaprdOLD

Step	Hyp	Ref	Expression
1		elmaprdOLD.3	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m 𝐴))
2		elmaprdOLD.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		elmaprdOLD.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4	2, 3	elmapd 8842	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m 𝐴) ↔ 𝐹:𝐴⟶𝐵))
5	1, 4	mpbid 235	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ⟶wf 6533  (class class class)co 7416   ↑_m cmap 8829

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8831

This theorem is used by: (None)